*DECK DSICO SUBROUTINE DSICO (A, LDA, N, KPVT, RCOND, Z) C***BEGIN PROLOGUE DSICO C***PURPOSE Factor a symmetric matrix by elimination with symmetric C pivoting and estimate the condition number of the matrix. C***LIBRARY SLATEC (LINPACK) C***CATEGORY D2B1A C***TYPE DOUBLE PRECISION (SSICO-S, DSICO-D, CHICO-C, CSICO-C) C***KEYWORDS CONDITION NUMBER, LINEAR ALGEBRA, LINPACK, C MATRIX FACTORIZATION, SYMMETRIC C***AUTHOR Moler, C. B., (U. of New Mexico) C***DESCRIPTION C C DSICO factors a double precision symmetric matrix by elimination C with symmetric pivoting and estimates the condition of the C matrix. C C If RCOND is not needed, DSIFA is slightly faster. C To solve A*X = B , follow DSICO by DSISL. C To compute INVERSE(A)*C , follow DSICO by DSISL. C To compute INVERSE(A) , follow DSICO by DSIDI. C To compute DETERMINANT(A) , follow DSICO by DSIDI. C To compute INERTIA(A), follow DSICO by DSIDI. C C On Entry C C A DOUBLE PRECISION(LDA, N) C the symmetric matrix to be factored. C Only the diagonal and upper triangle are used. C C LDA INTEGER C the leading dimension of the array A . C C N INTEGER C the order of the matrix A . C C Output C C A a block diagonal matrix and the multipliers which C were used to obtain it. C The factorization can be written A = U*D*TRANS(U) C where U is a product of permutation and unit C upper triangular matrices, TRANS(U) is the C transpose of U , and D is block diagonal C with 1 by 1 and 2 by 2 blocks. C C KPVT INTEGER(N) C an integer vector of pivot indices. C C RCOND DOUBLE PRECISION C an estimate of the reciprocal condition of A . C For the system A*X = B , relative perturbations C in A and B of size EPSILON may cause C relative perturbations in X of size EPSILON/RCOND . C If RCOND is so small that the logical expression C 1.0 + RCOND .EQ. 1.0 C is true, then A may be singular to working C precision. In particular, RCOND is zero if C exact singularity is detected or the estimate C underflows. C C Z DOUBLE PRECISION(N) C a work vector whose contents are usually unimportant. C If A is close to a singular matrix, then Z is C an approximate null vector in the sense that C NORM(A*Z) = RCOND*NORM(A)*NORM(Z) . C C***REFERENCES J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W. C Stewart, LINPACK Users' Guide, SIAM, 1979. C***ROUTINES CALLED DASUM, DAXPY, DDOT, DSCAL, DSIFA C***REVISION HISTORY (YYMMDD) C 780814 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890831 Modified array declarations. (WRB) C 891107 Modified routine equivalence list. (WRB) C 891107 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C 900326 Removed duplicate information from DESCRIPTION section. C (WRB) C 920501 Reformatted the REFERENCES section. (WRB) C***END PROLOGUE DSICO INTEGER LDA,N,KPVT(*) DOUBLE PRECISION A(LDA,*),Z(*) DOUBLE PRECISION RCOND C DOUBLE PRECISION AK,AKM1,BK,BKM1,DDOT,DENOM,EK,T DOUBLE PRECISION ANORM,S,DASUM,YNORM INTEGER I,INFO,J,JM1,K,KP,KPS,KS C C FIND NORM OF A USING ONLY UPPER HALF C C***FIRST EXECUTABLE STATEMENT DSICO DO 30 J = 1, N Z(J) = DASUM(J,A(1,J),1) JM1 = J - 1 IF (JM1 .LT. 1) GO TO 20 DO 10 I = 1, JM1 Z(I) = Z(I) + ABS(A(I,J)) 10 CONTINUE 20 CONTINUE 30 CONTINUE ANORM = 0.0D0 DO 40 J = 1, N ANORM = MAX(ANORM,Z(J)) 40 CONTINUE C C FACTOR C CALL DSIFA(A,LDA,N,KPVT,INFO) C C RCOND = 1/(NORM(A)*(ESTIMATE OF NORM(INVERSE(A)))) . C ESTIMATE = NORM(Z)/NORM(Y) WHERE A*Z = Y AND A*Y = E . C THE COMPONENTS OF E ARE CHOSEN TO CAUSE MAXIMUM LOCAL C GROWTH IN THE ELEMENTS OF W WHERE U*D*W = E . C THE VECTORS ARE FREQUENTLY RESCALED TO AVOID OVERFLOW. C C SOLVE U*D*W = E C EK = 1.0D0 DO 50 J = 1, N Z(J) = 0.0D0 50 CONTINUE K = N 60 IF (K .EQ. 0) GO TO 120 KS = 1 IF (KPVT(K) .LT. 0) KS = 2 KP = ABS(KPVT(K)) KPS = K + 1 - KS IF (KP .EQ. KPS) GO TO 70 T = Z(KPS) Z(KPS) = Z(KP) Z(KP) = T 70 CONTINUE IF (Z(K) .NE. 0.0D0) EK = SIGN(EK,Z(K)) Z(K) = Z(K) + EK CALL DAXPY(K-KS,Z(K),A(1,K),1,Z(1),1) IF (KS .EQ. 1) GO TO 80 IF (Z(K-1) .NE. 0.0D0) EK = SIGN(EK,Z(K-1)) Z(K-1) = Z(K-1) + EK CALL DAXPY(K-KS,Z(K-1),A(1,K-1),1,Z(1),1) 80 CONTINUE IF (KS .EQ. 2) GO TO 100 IF (ABS(Z(K)) .LE. ABS(A(K,K))) GO TO 90 S = ABS(A(K,K))/ABS(Z(K)) CALL DSCAL(N,S,Z,1) EK = S*EK 90 CONTINUE IF (A(K,K) .NE. 0.0D0) Z(K) = Z(K)/A(K,K) IF (A(K,K) .EQ. 0.0D0) Z(K) = 1.0D0 GO TO 110 100 CONTINUE AK = A(K,K)/A(K-1,K) AKM1 = A(K-1,K-1)/A(K-1,K) BK = Z(K)/A(K-1,K) BKM1 = Z(K-1)/A(K-1,K) DENOM = AK*AKM1 - 1.0D0 Z(K) = (AKM1*BK - BKM1)/DENOM Z(K-1) = (AK*BKM1 - BK)/DENOM 110 CONTINUE K = K - KS GO TO 60 120 CONTINUE S = 1.0D0/DASUM(N,Z,1) CALL DSCAL(N,S,Z,1) C C SOLVE TRANS(U)*Y = W C K = 1 130 IF (K .GT. N) GO TO 160 KS = 1 IF (KPVT(K) .LT. 0) KS = 2 IF (K .EQ. 1) GO TO 150 Z(K) = Z(K) + DDOT(K-1,A(1,K),1,Z(1),1) IF (KS .EQ. 2) 1 Z(K+1) = Z(K+1) + DDOT(K-1,A(1,K+1),1,Z(1),1) KP = ABS(KPVT(K)) IF (KP .EQ. K) GO TO 140 T = Z(K) Z(K) = Z(KP) Z(KP) = T 140 CONTINUE 150 CONTINUE K = K + KS GO TO 130 160 CONTINUE S = 1.0D0/DASUM(N,Z,1) CALL DSCAL(N,S,Z,1) C YNORM = 1.0D0 C C SOLVE U*D*V = Y C K = N 170 IF (K .EQ. 0) GO TO 230 KS = 1 IF (KPVT(K) .LT. 0) KS = 2 IF (K .EQ. KS) GO TO 190 KP = ABS(KPVT(K)) KPS = K + 1 - KS IF (KP .EQ. KPS) GO TO 180 T = Z(KPS) Z(KPS) = Z(KP) Z(KP) = T 180 CONTINUE CALL DAXPY(K-KS,Z(K),A(1,K),1,Z(1),1) IF (KS .EQ. 2) CALL DAXPY(K-KS,Z(K-1),A(1,K-1),1,Z(1),1) 190 CONTINUE IF (KS .EQ. 2) GO TO 210 IF (ABS(Z(K)) .LE. ABS(A(K,K))) GO TO 200 S = ABS(A(K,K))/ABS(Z(K)) CALL DSCAL(N,S,Z,1) YNORM = S*YNORM 200 CONTINUE IF (A(K,K) .NE. 0.0D0) Z(K) = Z(K)/A(K,K) IF (A(K,K) .EQ. 0.0D0) Z(K) = 1.0D0 GO TO 220 210 CONTINUE AK = A(K,K)/A(K-1,K) AKM1 = A(K-1,K-1)/A(K-1,K) BK = Z(K)/A(K-1,K) BKM1 = Z(K-1)/A(K-1,K) DENOM = AK*AKM1 - 1.0D0 Z(K) = (AKM1*BK - BKM1)/DENOM Z(K-1) = (AK*BKM1 - BK)/DENOM 220 CONTINUE K = K - KS GO TO 170 230 CONTINUE S = 1.0D0/DASUM(N,Z,1) CALL DSCAL(N,S,Z,1) YNORM = S*YNORM C C SOLVE TRANS(U)*Z = V C K = 1 240 IF (K .GT. N) GO TO 270 KS = 1 IF (KPVT(K) .LT. 0) KS = 2 IF (K .EQ. 1) GO TO 260 Z(K) = Z(K) + DDOT(K-1,A(1,K),1,Z(1),1) IF (KS .EQ. 2) 1 Z(K+1) = Z(K+1) + DDOT(K-1,A(1,K+1),1,Z(1),1) KP = ABS(KPVT(K)) IF (KP .EQ. K) GO TO 250 T = Z(K) Z(K) = Z(KP) Z(KP) = T 250 CONTINUE 260 CONTINUE K = K + KS GO TO 240 270 CONTINUE C MAKE ZNORM = 1.0 S = 1.0D0/DASUM(N,Z,1) CALL DSCAL(N,S,Z,1) YNORM = S*YNORM C IF (ANORM .NE. 0.0D0) RCOND = YNORM/ANORM IF (ANORM .EQ. 0.0D0) RCOND = 0.0D0 RETURN END